发表机构
Tehran Institute for Advanced Studies (TeIAS), Khatam University; University of Augsburg; Sharif University of Technology（德黑兰高等研究院（TeIAS），哈塔姆大学; 奥格斯堡大学; 谢里夫理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究无限制最大-最小度量T-join问题，证明UGC下3/2近似紧性，并针对精确基数变体提出基于层状割打包与树动态规划的确定性多项式时间近似算法，实现分段最优因子。
AI 中文摘要
在无限制的最大-最小度量$T$-join问题中，目标是寻找一个偶数终端集$T$，使得最小$T$-join的成本最大化。Iwata和Ravi给出了该问题的因子为$3/2$的近似算法。我们证明在唯一博弈猜想（UGC）下该保证是紧的：在UGC下，不存在因子严格小于$3/2$的多项式时间近似算法。然后我们考虑精确基数变体，该变体规定终端数为偶数$k$。记$p:=k/n$，我们为每个可行基数给出一个确定性的多项式时间$\rho(p)$-近似算法，其中$\rho(p)$分段定义为：当$0<p\ue2c2\ue2c27$时，$\rho(p)=7/2$；当$2/7\ue2c2p\ue2c22/3$时，$\rho(p)=1/p$；当$2/3\ue2c2p\ue2c27/8$时，$\rho(p)=1/[2(1-p)]$；当$7/8\ue2c2p<1$时，$\rho(p)=4$。特别地，在整个可行基数范围内，因子为$4$的近似始终成立；当$0<k\ue2c26n/7$时，因子至多为$7/2$；当$k=2n/3$时，因子等于$3/2$。该算法框架基于最优层状割打包、其加权树表示、精确基数舍入以及树动态规划。
英文摘要
In the unrestricted max--min metric $T$-join problem, one seeks an even terminal set $T$ maximizing the cost of a minimum $T$-join. Iwata and Ravi gave a factor-$3/2$ approximation for this problem. We show that this guarantee is tight under the Unique Games Conjecture: no polynomial-time approximation with factor strictly smaller than $3/2$ exists under UGC. We then consider the exact-cardinality variant, which prescribes an even number \(k\) of terminals. Writing \(p:=k/n\), we give a deterministic polynomial-time \(ρ(p)\)-approximation for every feasible cardinality, where \[ ρ(p)= \begin{cases} 7/2, & \makebox[1.5em][r]{$0$}<p\le2/7,\\ 1/p, & 2/7\le p\le2/3,\\ 1/[2(1-p)], & 2/3\le p\le7/8,\\ 4, & 7/8\le p<1. \end{cases} \] In particular, a factor-\(4\) approximation holds throughout the entire feasible cardinality range, the factor is at most \(7/2\) whenever \(0<k\le 6n/7\), and equals \(3/2\) at \(k=2n/3\). The algorithmic framework is based on optimal laminar cut packings, their weighted tree representations, exact-cardinality rounding, and tree dynamic programming.
